Buy-Back Contract as an Options Contract: A Comparative Analysis

To prove that a buy-back contract can be viewed as a special type of options contract, we need to show that we can construct an option contract with special parameters such that the sales as well as the costs are the same under both contracts.

Let's consider a buy-back contract where the manufacturer agrees to buy back any unsold units from the retailer at a predetermined price. Let's assume that the manufacturer sells the units to the retailer at a wholesale price of $w$ per unit, and agrees to buy back any unsold units at a price of $b$ per unit. Let's also assume that the retailer sells the units to the end consumers at a price of $p$ per unit.

Now, let's construct an option contract with the following parameters:

• Strike price: $s = w - b$
• Premium: $pr = 0$
• Expiration date: end of the selling season

Under this option contract, the retailer has the option to sell back any unsold units to the manufacturer at a price of $w - (w - b) = b$ per unit. However, the premium for this option contract is zero, which means that the retailer doesn't have to pay anything upfront to obtain this option.

Now, let's compare the costs and revenues under the buy-back contract and the option contract. If the retailer sells all the units to the end consumers, then there are no unsold units, and the costs and revenues are the same under both contracts. However, if there are unsold units, then the retailer can exercise the option to sell them back to the manufacturer at a price of $b$ per unit, which is the same as the buy-back price. This means that the costs and revenues are the same under both contracts.

Therefore, we have shown that a buy-back contract can be viewed as a special type of options contract with the above parameters.